The Existence and Averaging Principle for Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

Abstract

In this paper, we obtain the existence and uniqueness theorem for solutions of Caputo-type fractional stochastic delay differential systems(FSDDSs) with Poisson jumps by utilizing the delayed perturbation of the Mittag–Leffler function. Moreover, by using the Burkholder–Davis–Gundy inequality, Doob’s martingale inequality, and Hölder inequality, we prove that the solution of the averaged FSDDSs converges to that of the standard FSDDSs in the sense of Lp. Some known results in the literature are extended.

A Note on Averaging Principles for Fractional Stochastic Differential Equations

“…However, it must be noted that the convergence rate is ε 2α−1−γ instead of ε 1−γ unless α = 1. In other words, the convergence rate is related to the fractional order; this point is ignored in the existing literature due to the misuse of the standard form of fractional stochastic differential equations [3][4][5][6][7]9,10,15,16].…”

“…Li and Wang [15] presented an averaging theorem for Caputo FSDDEs driven by Gaussian white noise. Bai et al [16] proved an averaging result for Caputo FSDDEs with Poisson noise. Overall, the studies of averaging principles for FSDEs gained rapid development with tremendous achievements.…”

Hyers–Ulam Stability of Caputo Fractional Stochastic Delay Differential Systems with Poisson Jumps

“…Li and Wang in [21] studied the existence and uniqueness of a class of Caputo fractional stochastic delay differential systems (FSDDSs). In [22], we extended the main results of [21]. Up to now, to the best of our knowledge, the HUS of solutions for fractional stochastic delay differential systems (FSDDSs) has not been investigated.…”

“…Up to now, to the best of our knowledge, the HUS of solutions for fractional stochastic delay differential systems (FSDDSs) has not been investigated. Motivated by [19][20][21][22], in the present paper, we study the Hyers-Ulam stability for the following Caputo FSDDSs with Poisson jumps:…”